Inversion of Mutually Orthogonal CA Luca Mariot, Alberto Leporati Bicocca Security Lab (BiSLab) Dipartimento di Informatica, Sistemistica e Comunicazione (DISCo) Università degli Studi Milano - Bicocca ACRI 2018 – Como, September 17-21, 2018

Euler’s 36 Officers Problem « A very curious question [...] revolves around arranging 36 officers to be drawn from 6 differ- ent ranks and also from 6 different regiments so that they are ranged in a square so that in each line (both horizontal and vertical) there are 6 officers of different ranks and different regiments. » L. Euler, Sur une nouvelle espèce de quarrés magiques , 1782 Luca Mariot Inversion of Mutually Orthogonal CA

Latin Squares Definition A Latin square of order N is a N × N matrix L such that every row and every column are permutations of [ N ] = { 1 , ··· , N } 1 3 4 2 4 2 1 3 2 4 3 1 3 1 2 4 Luca Mariot Inversion of Mutually Orthogonal CA

Orthogonal Latin Squares (OLS) Definition Two Latin squares L 1 and L 2 of order N are orthogonal if their superposition yields all the pairs ( x , y ) ∈ [ N ] × [ N ] . 1 3 4 2 1 4 2 3 1,1 3,4 4,2 2,3 4 2 1 3 3 2 4 1 4,3 2,2 1,4 3,1 2,4 4,1 3,3 1,2 2 4 3 1 4 1 3 2 3,2 1,3 2,4 4,1 3 1 2 4 2 3 4 1 (a) L 1 (b) L 2 (c) ( L 1 , L 2 ) A set of n pairwise orthogonal Latin squares is denoted as n -MOLS Luca Mariot Inversion of Mutually Orthogonal CA

Secret Sharing Schemes (SSS) ( k , n ) Threshold Secret Sharing Scheme: a procedure enabling a dealer to share a secret S among n players so that at least k players out of n can recover S [Shamir79]. Example: ( 2 , 3 ) –scheme Setup Recovery B 1 P 1 P 1 B 1 S = B 2 P 2 P 2 B 2 B 3 P 3 P 3 B 3 Remark: ( 2 , n ) –scheme ⇔ set of n -MOLS Luca Mariot Inversion of Mutually Orthogonal CA

One-Dimensional Cellular Automata (CA) Definition One-dimensional CA: triple � m , n , f � where n ∈ N is the number of cells on a one-dimensional array, n ∈ N is the neighborhood and f : { 0 , 1 } n → { 0 , 1 } is the local rule. 0 00 Example: f ( x 1 , x 2 , x 3 ) = x 1 ⊕ x 2 ⊕ x 3 (Rule 150) 1 1 0 1 0 0 0 0 1 0 1 10 01 1 f ( 1 , 0 , 0 ) = 1 1 0 0 1 1 0 0 0 11 1 Luca Mariot Inversion of Mutually Orthogonal CA

Latin Squares through Bipermutive CA (1/2) ◮ Idea: determine which CA induce orthogonal Latin squares ◮ Bipermutive CA: local rule f is defined as f ( x 1 , ··· , x n ) = x 1 ⊕ ϕ ( x 2 , ··· , x n − 1 ) ⊕ x n ◮ ϕ : { 0 , 1 } n − 2 → { 0 , 1 } : generating function of f Lemma ([Eloranta93, Mariot16]) Let � 2 ( n − 1 ) , n , f � be a CA with bipermutive rule. Then, the global rule F generates a Latin square of order N = 2 n − 1 y n − 1 n − 1 y x x L ( x , y ) L ( x , y ) n − 1 Luca Mariot Inversion of Mutually Orthogonal CA

Latin Squares through Bipermutive CA (2/2) ◮ Example: CA � 4 , 1 , f � , f ( x 1 , x 2 , x 3 ) = x 1 ⊕ x 2 ⊕ x 3 (Rule 150) ◮ Encoding: 00 �→ 1 , 10 �→ 2 , 01 �→ 3 , 11 �→ 4 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 4 3 2 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 2 3 4 1 1 0 0 1 1 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 4 1 2 3 1 1 0 0 1 0 0 1 1 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1 3 2 1 4 0 1 1 0 0 0 1 1 (b) Latin square L 150 (a) Rule 150 on 4 bits Mutually Orthogonal Cellular Automata (MOCA): set of n bipermutive CA generating n -MOLS Luca Mariot Inversion of Mutually Orthogonal CA

MOCA by Linear CA ◮ Bipermutive Linear rule: f ( x ) = x 1 ⊕ a 2 x 2 ⊕···⊕ a n − 1 x n − 1 ⊕ x n ◮ Associated polynomial: f �→ P f ( X ) = a 1 + a 2 X + ··· + a n X n − 1 Theorem ([Mariot16]) A set of bipermutive linear CA are MOCA if and only if their associated polynomials are pairwise coprime 1,1 4,2 3,3 2,4 1 4 3 2 1 2 3 4 2,2 3,1 4,4 1,3 2 3 4 1 2 1 4 3 3 3 4,3 1,4 2,1 3,2 4 1 2 4 1 2 3 2 1 4 4 3 2 1 3,4 2,3 1,2 4,1 (a) Rule 150 (b) Rule 90 (c) Superposition Figure: P 150 ( X ) = 1 + X + X 2 , P 90 ( X ) = 1 + X 2 (coprime) Luca Mariot Inversion of Mutually Orthogonal CA

Inversion Problem in OCA ◮ Input : A pair w , z ∈ { 0 , 1 } n − 1 of final configurations ◮ Output : The unique preimage x generating w , z under the action of two OCA ↓ 1,1 4,2 3,3 2,4 ? ? ? ? 1 1 0 0 2,2 3,1 4,4 1,3 0 1 0 1 4,3 1,4 2,1 3,2 1 1 1 1 3,4 2,3 1,2 4,1 → (b) Input (c) Output (a) rule 90-150 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 100 1 1 1 , 1 1 , 1 010 0 1 0 , 0 110 1 0 10 01 001 1 1 0 , 1 101 0 0 1 , 0 1 , 0 011 1 0 11 111 0 1 0 , 1 ? ? ? ? 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 1 , 1 1 , 1 100 1 1 010 0 1 0 , 0 10 01 110 1 0 001 1 1 0 , 1 1 , 0 1 , 0 101 0 0 11 011 1 0 111 0 1 0 , 1 ? ? ? ? 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 1 , 1 1 , 1 100 1 1 010 0 1 0 , 0 10 01 110 1 0 0 , 1 001 1 1 1 , 0 1 , 0 101 0 0 11 011 1 0 111 0 1 0 , 1 1 1 ? ? 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 1 , 1 1 , 1 100 1 1 010 0 1 0 , 0 10 01 110 1 0 0 , 1 001 1 1 1 , 0 101 0 0 1 , 0 11 011 1 0 111 0 1 0 , 1 1 1 ? ? 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 100 1 1 1 , 1 1 , 1 010 0 1 0 , 0 110 1 0 10 01 001 1 1 0 , 1 101 0 0 1 , 0 1 , 0 011 1 0 11 111 0 1 0 , 1 0 1 ? ? 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Coupled De Bruijn Graph Idea : Walk on the De Bruijn graph labelled with both rules until a matching path is found. 0 , 0 ( x 1 , x 2 , x 3 ) f 90 f 150 000 0 0 00 1 , 1 100 1 1 1 , 1 010 0 1 0 , 0 110 1 0 10 01 001 1 1 0 , 1 101 0 0 1 , 0 1 , 0 011 1 0 11 111 0 1 0 , 1 0 1 0 0 0 1 1 1 Luca Mariot Inversion of Mutually Orthogonal CA

Inversion Algorithm I nvert -OCA ( G DB ( f , g ) , w , z ) V := V ertex ( G DB ( f , g )) E := E dges ( G DB ( f , g )) l := L abels ( G DB ( f , g )) c := NIL while e ∈ { ( v 1 , v 2 ) ∈ E : l ( v 1 , v 2 ) = ( w 1 , z 1 ) } AND c = NIL do c := DFS-M od ( V , E , l , v 1 , w , z ) end while return c Theorem Given two OCA rules f , g : { 0 , 1 } n → { 0 , 1 } and two final configurations w , z ∈ { 0 , 1 } n − 1 , algorithm I nvert -OCA returns the preimage x ∈ { 0 , 1 } 2 ( n − 1 ) of w , z in O ( n · 2 n ) steps Luca Mariot Inversion of Mutually Orthogonal CA

Conclusions and Future Directions Summing up: ◮ We considered the problem of inverting a pair of final configurations under the action of two OCA ◮ We devised an algorithm which solves the problem in exponential time wrt the CA diameter (but can be brought down to linear with parallelization!) Future directions: ◮ Design a cheater-immune SSS based on I nv -O ca ◮ Apply Genetic Programming (GP) to evolve MOCA with compact representation Luca Mariot Inversion of Mutually Orthogonal CA

References [delRey05] del Rey, Á.M., Mateus, J.P ., Sánchez, G.R.: A secret sharing scheme based on cellular automata. Appl. Math. Comput. 170(2), 1356–1364 (2005) [Eloranta93] Eloranta, K.: Partially Permutive Cellular Automata. Nonlinearity 6(6), 1009–1023 (1993) [Mariot17] Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary Algorithms for the Design of Orthogonal Latin Squares based on Cellular Automata. In: Proceedings of GECCO’17 (2017) [Mariot16] Mariot, L., Formenti, E., Leporati, A.: Construting Orthogonal Latin Squares from Linear Cellular Automata. In: Exploratory papers of AUTOMATA 2016 (2016) [Mariot14] Mariot, L., Leporati, A.: Sharing Secrets by Computing Preimages of Bipermutive Cellular Automata. In: Proceedings of ACRI 2014. LNCS vol. 8751, pp. 417–426. Springer (2014) [Shamir79] Shamir, A.: How to share a secret. Commun. ACM 22(11):612–613 (1979) [Tompa88] Tompa, M., Woll, H.: How to share a secret with cheaters. J. Cryptology 1(2), 133–138 (1988) Luca Mariot Inversion of Mutually Orthogonal CA

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